The diagonal of a square is 'a' cm, the area of square is :

Question

Accepted Answer

To determine the area of a square when the diagonal is given as 'a' cm, we need to use the relationship between the side of the square and its diagonal.

1. **Explanation for Option 3: $\frac{a^2}{2} \ \text{cm}^2$**

The diagonal of a square divides it into two right-angled triangles. If we let the side of the square be 's' cm, then by the Pythagorean theorem, we have:
   $$
   s^2 + s^2 = a^2
   $$
   Simplifying, we get:
   $$
   2s^2 = a^2
   $$
   Therefore, the side of the square is:
   $$
   s = \frac{a}{\sqrt{2}}
   $$
   The area of the square is given by the square of its side:
   $$
   \text{Area} = s^2 = \left(\frac{a}{\sqrt{2}}\right)^2 = \frac{a^2}{2} \ \text{cm}^2
   $$
   Thus, Option 3 is correct.

2. **Explanation for why other options are incorrect:**

- **Option 1: $2a \ \text{cm}^2$**

This option suggests that the area is directly proportional to the diagonal, which is incorrect. The area of a square is related to the square of its side, not directly to the diagonal.

- **Option 2: $\frac{a}{\sqrt{2}} \ \text{cm}^2$**

This option incorrectly suggests that the area is a linear function of the diagonal divided by $\sqrt{2}$. However, this expression actually represents the side length of the square, not the area.

- **Option 4: $\frac{a^2}{4} \ \text{cm}^2$**

This option suggests that the area is a quarter of the square of the diagonal, which is incorrect. The correct relationship, as shown, is that the area is half of the square of the diagonal.

In conclusion, Option 3 is the correct answer because it accurately represents the area of the square in terms of its diagonal using the derived formula.